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%TCIDATA{Created=Tue Dec 08 15:19:27 1998}
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\title{Synchronous oscillation in the cerebral cortex and object coherence:
simulation of basic electrophysiological findings.}
\author{J.J. Wright, P.D. Bourke, C.L. Chapman \\
%EndAName
{\small Mental Health Research Institute of Victoria, 155 Oak Street,
Parkville, }\\
{\small Melbourne, Victoria 3052, Australia. }\\
{\small Fax 61 3 9387 5061. }\\
{\small Email jjw@mhri.edu.au}}
\maketitle

\textbf{Abstract: }A lumped continuum model for electrocortical activity was
used to simulate several established experimental findings of synchronous
oscillation which have not all been previously embodied in a single
explanatory model. Moving-bar visual stimuli of different extension, stimuli
moving in different directions, the impact of non-specific cortical
activation upon synchronous oscillation, and the frequency content of EEG
associated with synchrony were considered. The magnitude of zero lag
synchrony was primarily accounted for by the properties of the eigenmodes of
the travelling LFP superposition waves generated by inputs to cortex,
largely independent of the oscillation properties and associated spectral
content. Approximation of the differences in cross correlation observed with
differently moving bar stimuli, and of the impact of cortical activation,
required added assumptions on (a) spatial coherence of afferent volleys
arising from parts of a single stimulus object, and (b) the presence of low
amplitude diffuse field noise, with enhancement of cortical signal/noise
ratio with respect to the spatially coherent inputs, at higher levels of
cortical activation. Synchrony appears to be a ubiquitous property of
cortex-like delay networks. Precision in the modelling of synchronous
oscillation findings will require detailed description of input pathways,
cortical connectivity, cortical stability, and aspects of
cortical/subcortical interactions.

\textbf{1.0 Introduction}

Discovery of the phenonomenon of synchronous oscillation in the cerebral
cortex [eg Eckhorn et al 1988; Gray et al 1989; Singer 1994; Singer and Gray
1995] was partly motivated by a proposal by von der Malsburg [1983] that
patterns of synchronous activity in the brain would offer a mechanism
whereby the many sensory input features contributing to the perception of a
unified object, might be linked to form a whole. This offered a solution to
the binding problem posed by the need to register unique combinations of all
possible stimulus features [Singer and Gray 1995; Livingstone 1996].

As has been pointed out by Palm and Wennekers [1997] interest in this
solution to the binding problem, rather than the simpler requirements of
direct modelling of experimental results has driven much of the attempts to
find an explanatory mechanism for synchronous oscillation. A strong
presumption that linked non-linear oscillations underlie the phenomenon [eg
Abarbanel et al 1996] has motivated most attempts at modelling the
phenomenon. Yet there is little evidence that phase-locking of nonlinear
oscillators is necessarily involved [Wennekers and Palm 1997], considering
the broad spectral band over which synchrony has been found [Bressler et al
1993].

The significance of the experimental data now published on synchronous
oscillation is very substantial, but remains controversial. Theoretical
opinions vary from the view that synchrony may be essentially irrelevant to
synaptic interactions [Amit 1998] to the possibility that synchrony is vital
to the co-ordination of synaptic modifications in the brain [Phillips and
Singer 1997]. Experimental conditions leave a number of variables
(connectivities, signal/noise ratios etc.) unspecified. Meantime, attempts
to model the process must be selective. Some experimental aspects of
synchronous oscillation appear reasonably clearly defined.

Firstly, it is known that synchrony is most often observed in association
with oscillation in the gamma band [eg Eckhorn 1988] but not uniquely so
[Bressler et al 1993].

Secondly, it appears that synchrony is strongest between cortical sites in
which neurones seem to be involved in co-processing of sensory input - eg,
between columns of cells with similar orientation preference [Gray and
Singer 1989; Livingstone 1996].

Thirdly, it appears increased cortical activation, such as is produced by
driving the mesencephalic reticular formation, enhances synchronous
oscillation [Munk et al 1996].

Fourthly, properties of the stimulus object(s) play a part. One large bar
moving across the visual field produces stronger synchrony than two small
bars concurrently stimulating each of the relevant points in the retina, and
bars moving in separate directions generally produce less synchrony than the
same bars moving in the same direction across the visual field
[Neuenschwander and Singer 1996; Livingstone 1996].

To attempt to provide a concise account for these experimental properties we
have used a lumped continuum model of the cortex which provides an account
of the general spectral content of EEG [Wright 1998]. This approach is
intended to complement other simulations which approximate physiological
realism using feedforward networks with inhibitory surrounds, or single and
multiple orientation domains [Schillen and Konig 1994; Xing and Gerstein
1996; Fuentes et al 1996; Juergens and Eckhorn 1997]. Our object was to
ascertain the minimal assumptions needed to reproduce the experimental data.
The model's parameterisation is as of yet approximate, and physiological
detail is incomplete, but these limitations are not relevant to the
demonstrations we will report. Models closely related to that applied here
have been analysed with regard to cross correlation properties under the
condition that two points on the cortical surface are driven by independent
white noise [Wright 1997; Robinson et al 1998; Chapman et al 1998] yielding
results which have guided the experimental design used here. In this model
synchrony depends wholly upon relatively long-range excitatory connections
in a continuum field - dendritic lag-summations and relatively rapid axonal
transmission being the essential ingredients - while short-range
excitatory/inhibitory interactions appear crucial only to the occurence of
oscillation. These characteristics are very similar to those observed in
neural network models with intracortical couplings only, observed by Wilson
and Bower [1991]. The continuum formulation supplements the neural network
approach, by enabling a different insight into the physical nature of
synchrony, since the essential nonlinearity of individual elements is
avoided, and the stochastic and essentially linear properties of the
neuronal mass is emphasised. Likewise, these intracortical two dimensional
models contrast with the work of Lumer et al [1997a,b] who studied
conditions for synchrony and oscillation in a neural-network model of the
thalamocortical system. They observed synchrony at many levels, which they
attributed largely to re-entrant activity at multiple levels in the pathway.

In the present model reproduction of the physiological experiments required
additional simple assumptions about the signal-to-noise ratio of cortex, and
cross correlation in afferent volleys associated with individual stimulus
objects, as will be described with the methods.

\textbf{2.0 Methods}

The simulation to be used here has been reported in Wright [1998].
Parameters were chosen to accord with the following assumptions about the
overall neuronal population properties:

(a) Most synapses are located on the distal dendritic tree, so mean delay
from synapses to soma is near the upper limit of physiological estimates for
dendritic delay. [Segev 1995; Thomson 1996, 1997]

(b) Dendritic delays in excitatory and inhibitory cells, and from excitatory
and inhibitory synapses, are comparable to first approximation.

(c) Axonal delays are range dependent, but are always small compared to
dendritic delays.

(d) There is finite probability that an action potential may be emitted by a
given neurone, even at membrane potentials very close to the inhibitory
reversal potential.

(e) Synaptic gains are in the ratio of 4:1 for inhibitory synapses versus
excitatory synapses [Segev 1995] and produce a high signal amplification
[Thomson 1997] so that the stable operating range of cortical activity is
restricted to low pulse densities.

\textbf{2.1 State-equations}

The $N$ cells in unit volume each have a probability of emission of an
action potential $q_i$ as a function of their membrane potentials. The sum
of population membrane potentials is taken to be directly proportional to
the LFP, $V(t)$ at time $t$. Then in a mean-field approximation the
pulse-probability density $Q(t)$ is given by

$Q=\frac 1N\stackrel{N}{\stackunder{i=1}{\sum }}q_i(V)$\hfill $(1)$

By the central limit theorem, for large $N$, $Q$ will have a Gaussian
distribution with respect to $V$, whatever the individual distributions of $%
q_i$, so $V$ and $Q$ are approximately related by\medskip

$Q=(1+e^{a(V-3)})^{-1}$\hfill $(2)$

This sigmoidal relation is used to approximate the sigmoidal error function
implied by the Gaussian population pulse probability distribution.Where $%
a=-\pi /\sqrt{3}$, LFP voltage units (vu) are approximate to standard
deviations of the distribution of cell pulse probability over the complete
range of LFP, with a 50\% mean probability of pulse emission 3sd from
complete polarisation of the neural population. Thus $Q$ has a value close
to zero when $V=0$, and approaching an asymptote of maximum pulse rate at $%
V=6$ vu.

The time response of mean membrane potential (and by implication LFP and
soma potential) is given by

$V(t)=g\stackrel{n}{\stackunder{j=1}{\sum }}w_jQ_a(t-j\Delta t)$ \ $%
j=1,2,3...n$\hfill $(3)$

where $g$ is synaptic gain, $Q_a$ is afferent pulse action density, $\Delta
t $ is the discrete time-step, and $n\Delta t$ is large compared to the peak
time response of membrane potential. In accord with Robinson et al [1997]

$w_j=b^2j\Delta te^{-bj\Delta t}$ $\hfill (3a)$

represents the rise and fall of membrane potential in response to input at $%
t=0$, incorporating lags due to both synaptic conduction and average
dendritic cable delay in a single function. Parameter $b$ regulates both the
peak time and mean delay associated with this lag. Time step $\Delta t$ was
set at 0.1ms, after trials showed that progressive decrements of time-step
to 0.01 ms produced only small, asymptotically diminishing effects on
spectral content of the results.

Within unit volumes both excitatory and inhibitory cell groups are
distinguished, each reciprocally and self-coupled, and each coupled at
longer range to other unit volumes by cortico-cortical fibres. Delays due to
axonal conduction between unit volumes are given by $\Delta \tau =r_{pq}/\nu
,$ where $\Delta \tau $ is axonal conduction lag over the distance $r_{pq}$
between the $p$th unit volume and the $q$th unit volume and axonal
conduction velocity is $\nu $.

Coupling strengths are proportional to:

(a) The fractional density of synaptic couplings afferent to the dendrites
of excitatory and inhibitory cells respectively ($\alpha _{ee},\beta
_{ei},\mu _{ei},M_{ee}$ etc., as listed in table 2).

(b) The synaptic gains of excitatory and inhibitory synapses, $g_e$ and $g_i$%
.

(c) Changes in synaptic efficacy, $E^{^{/}}$, representing feedback effects
including those of reversal potentials [Kandel and Schwartz 1985]. That is,
with increasing depolarization of cell membranes there is an increase in
sensitivity to inhibitory synaptic inputs and a decrease in sensitivity to
excitatory synapses. These feedback relations are modelled as linear
regressions of efficacy with membrane potential

$E_{ee}^{^{/}}(t)=(1-V_{e(p)}(t-\Delta t)/V_{eR})$\hfill $(4)$

$E_{ei}^{/}(t)=(1-V_{i(p)}(t-\Delta t)/V_{eR})$

$E_{ie}^{/}(t)=(1-V_{e(p)}(t-\Delta t)/V_{iR})$

$E_{ii}^{\prime }(t)=(1-V_{i(p)}(t-\Delta t)/V_{iR})$

where the subscripts $e$ and $i$ indicate excitatory and inhibitory
potentials, and subscript $R$ a constant-valued reversal potential.
Efficacies $\{E^{/}\}$ were applied with smoothing, so that in each case $%
E(t)=\sum_{j=1}^nu_jE^{/}(t-j\Delta t)$ where $u_j=ce^{-cj\Delta t}$,
describing an exponential decay of the impact of instantaneous membrane
potential upon synaptic efficacy. For high values of $c$ this decay is rapid
as would be expected for reversal potentials alone.

State equations for the cortical system are then given for the $p$th unit
volume by\medskip

$Q_{e(p)}=(1+e^{a(V_{e(p)}-3)})^{-1}$

$Q_{i(p)}=(1+e^{a(V_{i(p)}-3)})^{-1}$\hfill $(5)$

$V_{e(p)}=\sum\limits_{j=1}^nw_jQ_{ae(p)}(t-j\Delta t)$

$V_{i(p)}=\,\sum\limits_{j=1}^nw_jQ_{ai(p)}(t-j\Delta t)\hfill (5a)$

where $Q_{ae(p)},Q_{ai(p)}$ are afferent synaptic action densities for
excitatory and inhibitory cells in the $p$th unit volume, receiving local
synaptic input at negligible axonal delay and delayed cortico-cortical
inputs from $q$th$\;$unit volumes at range $r_{pq}$, $q=1...u.$in accord
with\medskip

$Q_{ae(p)}=g_e\beta _{ee}E_{ee(p)}Q_{e(p)}-g_i\beta _{ie}E_{ie(p)}Q_{i(p)}$

$+\,g_eM_{ee}E_{ee(p)}Q_{s(p)}+g_e\mu _{ee}E_{ee(p)}Q_{ns(p)}$

$+g_e\sum\limits_1^u\alpha _{ee}(r_{pq})E_{ee(p)}Q_{e(q)}(t-r_{pq}/\nu
)\hfill (5b)\medskip $

$Q_{ai(p)}=g_e\beta _{ei}E_{ei(p)}Q_{e(p)}-g_i\beta _{ii}E_{ii(p)}Q_{i(p)}$

$+\,g_eM_{ei}E_{ei(p)}Q_{s(p)}+g_e\mu _{ei}E_{ei(p)}Q_{ns(p)}$

$+g_e\sum\limits_1^u\alpha _{ei}(r_{pq})E_{ei(p)}Q_{e(q)}(t-r_{pq}/\nu
)\hfill (5c)$

where $\alpha _{ee}(r_{pq})$ and $\alpha _{ei}(r_{pq})$ are partial input
synaptic densities, such that $\stackrel{u}{\stackunder{1}{\sum }}\alpha
_{ee}(r_{pq})=\alpha _{ee}$ and $\stackrel{u}{\stackunder{1}{\sum }}\alpha
_{ei}(r_{pq})=\alpha _{ei}$. \medskip $Q_s$ and $Q_{ns}$ are system inputs. $%
Q_s$ represents all time-varying components in specific cortical afferents
and $Q_{ns}$ is a uniform DC input representing nonspecific cortical
activation, which acts as control parameter.

\textbf{2.2 Standard parameters and definition of units.}

State variables and parameters, their dimensions and standard values are
given in Tables 1 and 2. Where possible physiologically accurate values have
been applied, and certain difficulties of parameterisation have been avoided
by the use of normalised units. For details see Wright [1998].

2.3 \textbf{Configuration of Simulation}

An extended area of cortex was simulated by unit volumes in a 20 X 20
matrix, each volume connected with its neighbors so that the coupling
strengths, $\alpha _{ee}(r_{pq})$ declined with $r_{pq}$ as a Gaussian
function with standard deviation of 4 distance units,where a distance unit
was the side of one cell of the 20 X 20 matrix. This approximates to
distribution of cortico-cortical fibres in the cat brain if the distance
unit is taken as about 0.9 mm. Boundary conditions were toroidal in all
simulations reported. The application of absorbing boundary conditions and
changes in matrix size were also studied and these changes did not affect
the results to be reported.

\textbf{2.4 Added assumptions}

For the results which follow, two further assumptions proved necessary.
Discussion of these assumptions is reserved to the Conclusion.

(1) A single object moving in the sensory field stimulates feature detectors
in primary sensory neurones so that over any short epoch, afferent volleys
in the sensory pathway are correlated at zero lag - ie, inputs from a single
stimulus object give rise to spatially coherent input at cortical level.
Distinct stimulus objects are thus uncorrelated with each other by
definition.

(2) Cortical signal-to-noise ratio rises with cortical activation - ie, the
spatially coherent component of inputs from stimulus objects is greater
relative to background noise at higher levels of cortical activation.

\textbf{2.5 Simulation inputs and outputs}

Two configurations were used in these studies.

\textit{(a) Stationary driving inputs.}

These simulations follow Wright [1997] Robinson et al [1998], and Chapman et
al [1998] and are here used to succinctly demonstrate the basic physical
mechanism by which synchronous oscillation appears in the simulation, as a
prelude to the moving bar studies.

Specific inputs, $Q_s$, imitated time-variation of a complex localised input
to selective sites in cortex. These were delivered as two independent
(asynchronous) time series of zero-mean white noise [Marsaglia and Zaman
1987] of small amplitude, input to the unit volumes situated at sites in one
row of the matrix of elements, and seperated by five intervening elements.
Input was received at both excitatory and inhibitory cell dendritic
junctions of the driven sites.

Non-specific inputs, $Q_{ns},$ imitated the action of reticular,
catecholaminergic and other diffuse inputs to cortex, and were delivered
uniformly to all unit volumes in the matrix as constant non-zero inputs,
throughout the duration of each simulation run.

Although both types of input are extremely simple compared to real cortical
input, results are sufficiently general for interpretation of results from
inputs of greater complexity, as will be discussed.

Outputs were recorded as $V_e(t)$ and $Q_e(t)$ from all other elements in
the field, and were used to calculate lagged cross correlations, and to
calculate the major principle components (spatial eigenmodes) of activity in
the simulated cortical field, after prior removal of the signals from the $%
Q_s$ - driven sites. Spatial eigenmodes were calculated using spatial
principal component analysis (PCA) [Preisendorfer 1988] to resolve the wave
activity generated in the simulation into equivalent modes of spatially
stationary synchronous activity.

Simulation runs began with $\{Q_e,Q_i\}$ all initially zero, and ran for one
second, which ensured steady-state (signal mean stationary) conditions were
achieved. Outputs were analyzed over the subsequent second, to obtain data
for cross correlations, and two seconds to obtain data for principal
component analysis - the results of both these measures differing negligibly
with further prolongation of the run time.

\textit{(b) Moving bars.}

The simulation of moving bars implicitly defines a single moving object as a
set of cross correlated inputs, more or less spatially contiguous.

The simulated bars, which are $Q_s$ inputs, moved across the simulated
cortical surface, while non-specific activation, $Q_{ns}$ was delivered
uniformly as before. In some experiments the $Q_{ns}$ inputs included
spatio-temporal white noise in addition to the usual DC component, as will
be reported. This diffuse noise was provided by input of white noise time
series initiated from different seed values to each element in the cortical
matrix, allowing a more realistic representation of cortical activation. The
rms amplitude of specific inputs was 0.01 units, and the point amplitude of
diffuse field noise is reported in the results in the same units.

Moving bars were orientated along the rows of the matrix, and moved up or
down the columns. The standard apparent velocity of movement of the
simulated cortical surface was 24.4mm/s, so that a complete single sweep of
the bars occurred in 0.8192 s. Results from these runs required ensemble
averaging to yield sufficient confidence on magnitude, although robust
synchrony was apparent in individual sweeps. Bar speed was varied in control
experiments.

Short bars, of 3mm length in cortical projection, and a long bar of 20mm
(the width of the matrix) were simulated, each with an apparent width of 1mm
(one cortical element). The long bar thus reached completely across the
simulated cortical surface, while the short bars were aligned so as to pass
close to the two reference sites subsequently used for cross correlation.
Usually bar movements were such that the reference sites were crossed
simultaneously by the two short bars, or by the length of the long bar,
whatever the direction of movement. This occurred when the reference sites
were in the tenth row of the matrix. A control condition was the case in
which the reference sites were located in the fifth row of the matrix - that
is, as far as possible from the zeroth and tenth rows of the matrix, the two
positions where oppositely moving bars pass each other. This exceptional
condition we called the "Remote Passing" condition, to contrast it to the
more general "Proximal Passing" condition.

In the experiments to be reported, the two short bars moved so as to
straddle both reference sites, without directly driving either. This is not
critical to the results obtained, and it is relevant only that the short
bars pass close to the reference sites.

To prevent stepping discontinuities, the forward movement of the bars was
smoothed in imitation of a 1mm wide bar moving continuously.

In both long and short bars, only every second element along the length of
the bar was driven, with a 50\% probability of receiving input at each time
step. In no case did the reference sites themselves receive direct specific (%
$Q_s$) inputs, as to do so confounds cross correlations to be made with
regard to field effects in the simulation with the direct content of the
inputs. Both zero-mean and non-zero mean $Q_s$ inputs were applied in
different simulation runs

Finally, the white noise time series used to drive any single bar were
arranged according to either of two conditions. In the "Synchronous bar"
condition all inputs to a single bar were synchronous (identical), while
they were asynchronous (uncorrelated) between separate bars. This condition
meets the definition of a single moving object, as mentioned above. In the
"Asynchronous bar" condition all inputs within, as well as between bars,
were uncorrelated.

\textbf{3. Results.}

\textbf{3.1 Stationary driving inputs}

Figure 1 shows the essential mechanism whereby synchronous oscillation
arises in these simulations, as reported in Wright [1997], Robinson et al
[1998] and Chapman et al [1998]. We wish to emphasise that zero-lag
synchrony is a universal attribute of continuum fields or neural networks
which have summing junctions with delay (dendrites) and couplings (axons)
with transmission delay which is small compared to dendritic delay.
Synchrony then occurs by summation of even (in phase) components in the
seperated input signals, and cancellation of odd (out of phase) components.
This effect does not require nonlinearity.

When twin uncorrelated signals are delivered to the cortical surface, a
pattern of zero-lag synchrony develops around the driving sites, as is shown
in the upper diagrams in figure 1. Similar results can be obtained for cross
correlations of any reference point close to either of the sites of white
noise input. The pattern of zero-lag synchrony has been shown to be
sensitive to inhomogeneities in the strength of connections between elements
of the matrix, whether these reflect structural or dynamic coupling gains.
Conversely, uniform multiplications of the connection strength throughout
the field have no effect on the pattern of synchrony. Similarly, uniform
increases in cortical activation ($Q_{ns}$) have little effect, so long as
the level of activation remains below a critical level at which limit cycles
develop. A highly nonlinear dynamic emerges at very high $Q_{ns}$ and this
condition was avoided throughout the current studies.

The middle diagrams in figure 1 show the first and second eigenmodes
revealed by PCA. The first eigenmode is similar in form to the field of
zero-lag maximum cross correlation, and consumes the majority of the
variance in the field. It can be shown [Chapman et al 1998] that the first
eigenmode follows even (in-phase) transients in the driving inputs and the
second eigenmode follows odd (anti-phase) transients in the same inputs. The
lower diagrams of figure 1 show schematically the way in which the summation
of even components in the surrounding field creates the first eigenmode,
which predominates over the second eigenmode - the form of the second mode
being attributable in part to cancellation of the odd signals spreading into
the field from the inputs. Predominance of the first mode is greatest in the
field near the driving sites - hence the field of zero-lag synchrony.

\textbf{3.2 Impact of concurrent noise and cortical activation - static case 
}

Figure 2 shows the cross correlations found between two sites within the
field of zero-lag synchrony generated by the two stationary sites of driving
with asynchronous noise, while the rms amplitude of input to the driving
sites was held constant and diffuse noise throughout the field was increased.

It can be seen that with increasing field noise the cross correlation
decreases as expected, but there is little effect on the cross correlations
obtained when the degree of cortical activation is changed four-fold. The
lack of sensitivity of cross correlation to $Q_{ns}$ alone is accounted for
by the fact that the first and second eigenmodes of field activity increase
concurrently in amplitude in response to the uniformly increasing dynamic
gain ($dQ/dV$), produced by increase in $Q_{ns}$. Such small increase in
cross correlation as does occur with increasing $Q_{ns}$, is manifest at
higher ``noise-to-signal'' ratios.

\textbf{3.3 Moving bars}

Figure 3a shows the simulation's reproduction of the findings of Gray et al
[1989] and Eckhorn et al [1988] for moving bars. It can be seen that LFP
time series, power spectra, and cross correlations closely approximate
experimental findings. Only a single-sweep LFP time series is shown here,
but superposition of LFP from repeated sweeps shows that they are not
phase-locked to the presentation of the stimulus, in the same manner as
noted experimentally by Eckhorn et al [1988]. The particular frequency
content of the LFP time series, which matches the predominance of power in
the gamma range typically seen in experiments is a consequence of the choice
of $Q_{ns}$ applied in these particular simulations. At much lower values of 
$Q_{ns}$ the spectral content resembles EEG at lower levels of activation -
yet the cross correlation values are little affected.

Notably, the highest zero lag cross correlation is seen for the single large
bar, next for two short bars moving in the same direction, and lowest for
the two short bars moving in opposite directions. The relative magnitude of
these cross correlations is essentially independent of $Q_{ns}$.

Figure 3b shows data from simulations similar to those shown in 3a except
that the "asynchronous bar" condition has been applied. Notably the long bar
now yields no higher cross-correlation than two bars moving in the same
direction.

Figure 3c uses conditions matched to those in 3b, and cross-correlations are
shown for both the standard "Proximate passing" and the "Remote passing"
conditions. It is seen that variation in the situation of the reference
sites relative to the position of passing has little effect on the degree of
cross-correlation, so long as the sites remain in a symmetric relation to
the tracks of passage of the bars.

In all permutations of the various conditions applied, we found that when
the diffuse field noise was zero, no differences could be demonstrated in
cross-correlation for two bars moving in the same, versus opposite,
direction. This result may appear paradoxical, since the two bars moving
together are always closer to each other, therefore interact more strongly,
and create larger amplitude travelling waves in the medium. However, this
greater wave amplitude does not lead to any significant difference in the
partition of energy between the first and second eigenmodes of the waves.
Thus cross-correlations (which normalise absolute magnitude) are not
sensitive to the relative direction of bar movement so long as the paths
travelled are otherwise identical.

The introduction of diffuse field noise produces sensitivity to relative
direction of movement of the bars, as shown in figure 3d. This result is
accounted for as follows: when the two bars move in the same direction the
absolute magnitude of both the first and second eigenmodes is enhanced,
although their relative magnitudes are unchanged. With equivalent levels of
additive field noise the signal/noise ratio - which is measured by
cross-correlation - is then higher for the case in which the bars move
together, and lower when they move in opposite directions.

\textbf{3.4 Effect of cortical activation with concurrent increase in
cortical signal-to-noise ratio - two moving bars.}

Figure 4 shows that so long as constant amplitude diffuse spatio-temporal
noise is applied to the field, then the zero-lag synchrony increases as
cortical activation is increased. These results are closely similar to the
experimental results reported in Munk et al [1996]

\textbf{3.5 Consequences of other alterations of signal properties}

Qualitatively, the results in figures 3 and 4 do not depend upon the input
signals associated with the moving bar having a non-zero mean. The results
shown are for zero-mean input signals. Non-zero mean of the moving bar
inputs inevitably adds to the cross-correlation observed, unless high-pass
filtering is applied to the output signals before cross correlation is
performed. (This offset or envelope effect is often removed experimentally
by the use of high pass filters to allow for envelope of the input signal.)
No input signal correlations are required for synchrony to emerge, no matter
what the size or direction of movement of the bars.

The results shown are also independent of the choice of bar speed, which was
doubled and halved from the standard speed without significant consequence
to results. This is not congruent with Eckhorn et al's [1988] findings that
increasing stimulus speed was associated with gamma-band oscillation at
increasing frequency, and implies that such a relation would require
stimulus speed to be associated with level of cortical activation.

With regard to figure 4, if the amplitude of the specific stimuli does not
increase along with cortical activation, or if there is no diffuse field
noise, then the maximum cross correlation does not alter significantly.

\textbf{4. Conclusions}

The above results appear to capture concurrently the physiologically
observed cross correlation results of moving bar experiments and influences
of reticular formation stimulation, along with realistic representation of
LFP\ spectral power, and absence of phase-locking of response to the input
signal.

The simulated cortical medium has the capacity to selectively eliminate odd
components in the asynchronous input signals. This basic property carries
over into more complex moving stimuli, is little effected by the level of
cortical activation per se - and correspondingly, does not require
oscillation in the gamma range for synchrony to occur.

The ubiquity of synchrony without dependence upon any correlation in the
inputs does not provide a sufficient explanation for either the experimental
differences dependent on the direction of bar movement, or the effects of
cortical activation upon synchrony. Reproduction of these physiological
results depends upon two additional assumptions - firstly, that a single
stimulus object can be defined as a spatially coherent set of inputs, and
secondly that the signal to noise ratio of inputs increases concurrently
with cortical activation.

In the results reported above, the first assumption proved necessary only to
account for the enhanced magnitude of cross-correlations found when the
stimulus is a long single bar, rather than two short bars moving in the same
direction. This is a robust finding, both physiologically and within our
simulations.

The first assumption may be usefully regarded as providing a definition of a
stimulus object. Extension of this definition leads to the notion that any
spatially synchronous pattern of activity in the brain is an ``object'' -
either the representation of a physical object via sensory input, or a
coherent pattern partly internally generated, and thus a representation of a
mental object. This is equivalent to von der Malsburg's [1983] original
proposition that synchronous oscillation is the means by which binding is
mediated. Concurrent stimulation of receptor cells in cortex or elsewhere
must produce synchrony in the afferent bombardment over at least some
frequencies as has been remarked in physiological experiments [eg
Neuenschwander and Singer 1996; Steriade et al 1996; Kreiter and Singer
1996] and demonstrated in models of the visual pathways [Ghose and Freeman
1997]. It is not necessary that coherence in the afferent volleys be
complete at all frequencies, to give rise to the effects reported.

The second assumption we have shown to account for experimentally observed
increases in cross-correlation with increasing cortical activation. More
generally, a degree of diffuse field noise seems essential to account for
sensitivity of cross-correlation to relative direction of movement of
multiple bars. There are implications here for the coherent infomax
hypothesis, regarding synaptic modification and learning in relation to
synchronous oscillation (Phillips and Singer 1997). The
information-theoretical derivation of coherent infomax leaves unexplained
how information from different stimulus configurations becomes selectively
distributed over synapses. Our findings suggest that noise may be required
to enhance contrasts of relative synaptic activation when stimulus objects
move in relation to each other in different ways.

A relation between cortical activation and signal/noise ratio is without
general proof from physiological data. It is inherently plausible, since
cortical activation is associated with enhanced cortical information
processing. It is loosely supported by the close association of pulse and
LFP in cortical activity in the gamma band, as opposed to the notoriously
low associations of EEG and pulse activity at lower EEG frequencies and
cortical activation [Stryker 1989]. A part of the enhancement of signal
versus noise may be attributed to a further property of synchronous
oscillation which is only weakly apparent in these simulations. As shown in
figure 2, increasing cortical activation does somewhat enhance cross
correlation at medium ranges of signal-to-noise. This effect would be
further enhanced were the effects of cortical depth or repeated passage of
signals through layers of relay cells such as the cell layers of the lateral
geniculate body included.

We were forced to make these assumptions because of the simplicity of
representation of the input pathways in our model. The model of Lumer et al
[1997a,b], in contrast, dealt in detail with the input pathways, and
reproduced synchronous oscillation as a general property of jitter stimuli
introduced to their neural networks. Similar considerations hold for the
earlier cited works on feedforward modelling [ Xing and Gerstein 1996;
Fuentes et al 1996; Schillen and Konig 1994; Juergens and Eckhorn 1997] It
would appear that these approachs and ours may be complementary.

One property of experimental synchronous oscillation which has not been
reproduced is the relation of bar speed to LFP spectral content [Eckhorn et
al 1988]. Within the present model, this could be explained by the further
assumption that faster moving stimuli contribute more strongly to cortical
activation, whether directly or via collaterals in the reticular activating
system. This would have the effect of shifting the LFP spectrum further to
the right, in accord with experimental observation. In a study of somatic
sensory synchrony by Ruiz et al [1995] faster stimuli were associated with
increased cell firing rates, consistent with an association between stimulus
velocity and total activation.

The present results do not include two further factors which may be crucial
to quantitative reproduction of synchronous oscillation, whereas we have
been able to obtain only the correct relative changes in cross-correlation
magnitude according to different experimental conditions.

Firstly, we have here considered only symmetrical cortical couplings and
uniform cortical activation, whereas in reality both these factors are
inhomogeneous, and systematically so. This may explain why our simulations
did not capture the extremes of cross correlation seen experimentally, in
which cross correlation may be sometimes reduced to zero for bars moving in
opposite directions, while quite high correlations are seen for movements in
the same direction. As earlier remarked there is other evidence in similar
simulations that nonuniform structural and dynamic coupling strengths change
the patterns of synchronisation for static inputs. Introduction of
nonuniform couplings would open the possibility of detailed simulation of
synchrony - for example, between cortical sites of similar orientation
preference - since such sites are more strongly coupled than sites of
dissimilar orientation preference [Malach et al 1993; Yoshioka et al 1996].
Like-to-like connectivity might also hold for velocity characteristics of
stimuli, as is predicted by certain Hebbian models for the development of
intracortical intrinsic connections [Alexander et al 1999].

Secondly, these results do not allow for the possible occurrence of
nonlinear phase locking as a mechanism of synchronous oscillation. At higher
levels of cortical activation nonlinear phase locking occurs in these
simulations, as reported in Chapman et al [1998]. This additional mechanism
opens the door to modelling of the putative role of synchrony in image
segmentation, as shown in abstract simulations by Wang [1996]. The form of
synchronisation modelled here acts as a means of rapid synchronisation of
cortical areas active for whatever reason - ie, the present mechanism would
appear sufficient to mediate binding.

Beyond these considerations of intracortical dynamics and input path
properties,still more complete accounts of synchronous oscillation will
likely require an account of cortical/subcortical interactions - processes
which appear required to organise the binding and uncoupling of cortical
assemblies during ongoing perception and cognition [Rodriguez et al 1999;
Miltner et al 1999].

Thus extensions of these experiments may be expected to enable testing of
the mechanisms proposed, against increasingly complicated physiological
experiments - at the price that, experimentally, concurrent levels of
cortical activation and the detailed local connectivity of cells must be
explicitly considered.

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\pagebreak

\begin{tabular}{|lll|}
\hline
$V_e$ & {\small excitatory LFP} & {\small vu} \\ 
$V_i$ & {\small inhibitory LFP} & {\small vu} \\ 
$Q_e$ & {\small excitatory PPD} & {\small dimensionless} \\ 
$Q_i$ & {\small inhibitory PPD} & {\small dimensionless} \\ 
$a$ & {\small slope parameter} & $-\pi /\sqrt{3}${\small \ (vu}$^{-1}$%
{\small )} \\ 
$b$ & {\small dendritic time-constant} & {\small 50s}$^{-1}$ \\ 
$g_e$ & {\small excitatory gain} & {\small 65 vu} \\ 
$g_i$ & {\small inhibitory gain} & {\small 260 vu} \\ 
$c$ & {\small decay time-constant.} & {\small 1000s}$^{-1}${\small .} \\ 
$\nu $ & {\small axonal velocity} & {\small 9 m/sec.} \\ 
$\sqrt{\stackrel{-}{r}_{pq}^2}$ & {\small sd of axonal range} & {\small 4mm.}
\\ 
$V_{eR}$ & {\small EPSP reversal} & {\small 12 vu} \\ 
$V_{iR}$ & {\small IPSP reversal} & {\small -0.02 vu} \\ 
$Q_{ns}$ & {\small nonspecific input} & {\small dimensionless} \\ 
$Q_s$ & {\small specific input} & {\small dimensionless} \\ \hline
\end{tabular}

{\small Table 1. State-variables and standard parameters other than synaptic
densities. LFP=local field potential, PPD=pulse probability density.\bigskip
\bigskip }

\pagebreak

\begin{tabular}{|l|l|l|}
\hline
{\small Synaptic } & {\small Synaptic} & {\small Afferent } \\ 
{\small coupling} & {\small density fraction} & {\small fraction.} \\ \hline
$\alpha _{ee}$ & $0.765$ & $0.8693$ \\ 
$\beta _{ee}$ & $0.0845$ & $0.0960$ \\ 
$\beta _{ei}$ & $0.0149$ & $0.1242$ \\ 
$\alpha _{ei}$ & $0.100$ & $0.8333$ \\ 
$\beta _{ie}$ & $0.0228$ & $0.0259$ \\ 
$\beta _{ii}$ & $0.004$ & $0.0333$ \\ 
$\mu _{ee}$ & $0.0077$ & $0.0088$ \\ 
$\mu _{ei}$ & $0.0011$ & $0.0092$ \\ 
$M_{ee,i}$ & {\small Not given.} &  \\ \hline
\end{tabular}

{\small Table 2. Synaptic couplings subscripts }$ee,$ $ei,$ $etc.,$ {\small %
indicate synapses between cell types, excitatory to excitatory, excitatory
to inhibitory, etc.}

{\small Types of coupling are: }$\alpha ${\small \ (cortico-cortical
connections), }$\beta ${\small \ (intracortical connections), }$\mu ${\small %
\ (nonspecific cortical afferents) and }$M${\small \ (specific afferents).}

{\small Synaptic density fraction is the proportion of synapses of each type
in unit cortical volume. (The exact values used in the simulations are given
for completeness [Liley and Wright 1994] although the precision given is
greater than is justified from the anatomical data.) Afferent fraction is
the proportion of synapses on the excitatory or inhibitory cell dendrites
respectively, and are thus the values applied in equations (5).\bigskip
\bigskip }\pagebreak 

{\small Figure 1. Essential properties of synchronous oscillation.\newline
Top figures: A representation of the simulated cortical surface. Open
squares represent the sites of input of uncorrelated white noise. The filled
square is the reference point from which cross-correlations are calculated
with respect to the rest of the field. Top left: maximum positive
cross-correlations. Top right: delay associated with maximum
cross-correlation.\newline
Middle figures: The first and second principal eigenmodes of activity on the
simulated cortical surface, obtained from the same data used to generate the
top figures..\newline
Bottom figures: Schematic ``freeze frame'' images of potentials (or pulse
densities) on the simulated cortical surface seen when the twin inputs are
sine waves, of any single frequency. Bottom left: potentials in the field
when the sine waves are in phase. Bottom right: potentials in the field when
the sine waves are of reverse phase.}\pagebreak 

{\small Figure 2. Dependency of cross-correlation on signal/noise ratio and
independence from uniform cortical activation in the absence of overlying
noise. Zero-lag cross correlation obtained between the reference site shown
in figure 1, and another site located at the point of mirror-image symmetry,
adjacent to the right hand input site. Mean and SEM of cross-correlations
from 20 1s simulation runs. In addition to the twin sites of asynchronous
noise input to the same driving sites as in figure 1, spatio-temporally
white noise has been input to all other sites in the field. The
noise-to-signal ratio refers to the ratio of rms amplitude of the random
field noise, to that at the twin ``driving'' sites. Results are shown for
two widely different conditions of non-specific cortical activation, as
indicated by the value of parameter }$Q_{ns}${\small .}\pagebreak 

{\small Figure 3a. Effects of stimulus extension and coherence on
synchronous oscillation.\newline
Oscillations of local field potential (left hand figures), spectral power
(middle figures) and pulse-density cross-correlations (right hand figures)
in the simulated cortical field, as zero-mean inputs are swept across the
field in imitation of the visual stimuli associated with moving bars. A
single bar has coherent white noise inputs applied at each portion of the
bar. Separate bars have no input signal correlations - see text for details.
Diffuse field noise amplitude 0.000033. Top graphs: two short bars moving in
the same direction. Middle graphs: two short bars moving in opposite
directions. Bottom graphs: one long bar moving across the field. In this
figure and in all following figures, cross-correlations were associated with
standard errors of approximately 0.005 +/- 0.001. Thus, visible differences
in cross-correlation are highly statistically significant. }

{\small Figure 3b. Results shown are derived as for figure 3a except that
uncorrelated white noise was delivered to all parts of each bar, as well as
being uncorrelated between bars. Diffuse field noise amplitude 0.000025. }

{\small Figure 3c. Cross-correlations for the asynchronous bar case, with
diffuse field noise 0.000025, for bars moving in the same and opposite
directions. Reference points at the matrix row at which the oppositely
moving bars pass each other (proximate pass) or in the same columns but
remote from the passing row (remote pass). See text for details. Differences
are between the remote and proximate conditions. The small differences
remain statistically significant. }

{\small Figure 3d. Cross-correlations from proximate-pass reference sites
with differing levels of diffuse field noise, for both asynchronous and
synchronous bar conditions. Differences are with respect to the directions
of bar movement shown in the graphs to the left of each difference graph.
The difference produced by direction of movement is zero at zero lag in the
absence of diffuse field noise. }\pagebreak $\theta $

{\small Figure 4. Effects of cortical activation on synchronous oscillation
in the presence of uniform field noise.\newline
Average power spectra and pulse-cross correlations calculated as for the two
short bars moving in the same direction case, in figure 3a. Here uniform
amplitude spatio-temporally white noise was applied throughout the field,
and both the level of cortical activation (}$Q_{ns}${\small ) and the
amplitude of the signals associated with the moving bars (}$Q_s${\small )
were raised concurrently.}

\end{document}
